Relations between the "$\forall$" quantifier and integration

Question

I'm studying introduction to mathematical logic, we recently began predicate logic and I find similarities between the "$\forall$" predicate and integration:
1. They both have a dummy variable, which they "enumerate" over its possible values
2. The fact that $\forall x \forall y \alpha \vDash \forall y \forall x \alpha$ is similar to Fubini's theorem.

Is there a formalism that relates the two?

Answer 1

I actually suspect there's a closer relation between integration and the existential quantifier.

From the perspective of category theory, summation is an instance of a coproduct, which is in turn closely related to existential quantification. For instance:

• Given a family of sets $(X_i \mid i \in I)$, their disjoint union $\sum_{i \in I} X_i$ satisfies the following universal property: functions $f : \sum_{i \in I} X_i \to Y$ correspond with $I$-indexed families of functions $f_i : X_i \to Y$. (More generally, replace 'sets' with 'objects of a category', 'functions' with 'morphisms', and 'disjoint union' with 'coproduct'.)
• Given a proposition $\varphi(x)$, where $x$ ranges over some set $I$, proofs of $[(\exists x \in I) \varphi(x)] \Rightarrow \psi$ correspond with proofs of $\varphi(i) \Rightarrow \psi$ for each $i \in I$.

[In fact, from the perspective of Martin-Löf dependent type theory, the notions of 'disjoint union of sets' and 'existentially quantified formula' are identified.]

Another perspective is that indexed disjoint unions and existential quantification are both instances of left adjoints to substitution functors of the appropriate kinds. I'll spare you the details now, but Sections 9.5 and 9.7 of Steve Awodey's textbook on category theory is a good starting point.

Now integration generalises summation and coends generalise coproducts, and these generalisations happen in similar ways (people smarter than myself have studied connections between integrals and coends).

So in conclusion, you're thinking along the right lines, but I think you'll find more parallels between integration and $\exists$ than integration and $\forall$.

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Addendum: this blog post by Bartosz Milewski discusses the connection between coends and the existential quantifier as implemented in Haskell. It says:

Just like the end is related to a product, the coend is related to a coproduct, or a sum (in this respect, it resembles an integral, which is a limit of a sum). Rather than having projections, we have injections going from the diagonal elements of the profunctor down to the coend. If it weren’t for the cowedge conditions, we could say that the coend of the profunctor p is either p a a, or p b b, or p c c, and so on. Or we could say that there exists such an a for which the coend is just the set p a a. The universal quantifier that we used in the definition of the end turns into an existential quantifier for the coend.

This is why, in pseudo-Haskell, we would define the coend as:

exists a. p a a

Answer 2

If you do the following substitutions: $$\begin{align} &+ \to \min, \\ & \times \to + \end{align}$$ in the definition of the Riemann integral $\int_a^b f(x)\,dx$, you get the equivalent of $\min_{x\in[a,b]} f(x)$. More on max-plus algebra.

In the event that $f$ has codomain $\{0,1\}$, this is the same as the indicator of $\forall x\in[a,b],f(x)=1$

More generally, operations like $\int$, $\sum$, $\max$ and $\forall$ merely sum a function $f(x)$ over a set. "Summation" in this case should be interpreted in an abstract way as a commutative monoid operation. And in the case of $\int$, the sum must be "scaled down" in order to not blow up.

Answer 3

What follows might be relevant to what you’re asking about. The notions $\exists^{*},$ $\forall^{*}$ (there exists non-meagerly many, for co-meagerly many) and $\exists_{\mu}^{*},$ $\forall_{\mu}^{*}$ (there exists non-$\mu$-measure zero many, for $\mu$-almost all) can be found on pages 53 and 114, respectively, of Kechris [1]. These generalized quantifier notions are additional examples of what I discussed in my answer to Why haven't mathematicians come up with an efficient way of writing “sufficiently”, e.g. “for $n$ sufficiently large”, although I didn’t bother mentioning them there because my answer was probably already too technical for the question.

Anyway, in [2], Lyons observes (Proposition 1) that Fubini’s Theorem for measure zero sets implies the quantifiers $\forall_{\mu}^{*}$ and $\forall_{\nu}^{*}$ commute (and hence by taking negations, $\exists_{\mu}^{*}$ and $\exists_{\nu}^{*}$ also commute), where $(X,\mu)$ and $(Y,\nu)$ are $\sigma$-finite measure spaces. More precisely, if $P(x,y)$ is a measurable predicate, meaning that the subset of $X \times Y$ where $P$ is true is $\mu \times \nu$ measurable, then $\forall_{\mu}^{*}\,\forall_{\nu}^{*}\,P(x,y)$ holds if and only if $\forall_{\nu}^{*}\,\forall_{\mu}^{*}\,P(x,y)$ holds.

[1] Alexander Sotirios Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics #156, Springer-Verlag, 1995, xviii + 402 pages.

[2] Russell David Lyons, Measure-theoretic quantifiers and Haar measure, Proceedings of the American Mathematical Society 86 #1 (September 1982), 67-70. [See also the non-trivial corrections in Erratum to Measure-theoretic quantifiers and Haar measure, Proc. AMS 91 #2 (June 1984), 329-330.]